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Theorem odngen 14987
Description: A cyclic subgroup of size  ( O `  A ) has  ( phi `  ( O `  A
) ) generators. (Contributed by Stefan O'Rear, 12-Sep-2015.)
Hypotheses
Ref Expression
odhash.x  |-  X  =  ( Base `  G
)
odhash.o  |-  O  =  ( od `  G
)
odhash.k  |-  K  =  (mrCls `  (SubGrp `  G
) )
Assertion
Ref Expression
odngen  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  e.  NN )  -> 
( # `  { x  e.  ( K `  { A } )  |  ( O `  x )  =  ( O `  A ) } )  =  ( phi `  ( O `  A ) ) )
Distinct variable groups:    x, A    x, G    x, K    x, O    x, X

Proof of Theorem odngen
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eqid 2358 . . . 4  |-  ( y  e.  ( 0..^ ( O `  A ) )  |->  ( y (.g `  G ) A ) )  =  ( y  e.  ( 0..^ ( O `  A ) )  |->  ( y (.g `  G ) A ) )
21mptpreima 5248 . . 3  |-  ( `' ( y  e.  ( 0..^ ( O `  A ) )  |->  ( y (.g `  G ) A ) ) " {
x  e.  ( K `
 { A }
)  |  ( O `
 x )  =  ( O `  A
) } )  =  { y  e.  ( 0..^ ( O `  A ) )  |  ( y (.g `  G
) A )  e. 
{ x  e.  ( K `  { A } )  |  ( O `  x )  =  ( O `  A ) } }
32fveq2i 5611 . 2  |-  ( # `  ( `' ( y  e.  ( 0..^ ( O `  A ) )  |->  ( y (.g `  G ) A ) ) " { x  e.  ( K `  { A } )  |  ( O `  x )  =  ( O `  A ) } ) )  =  ( # `  { y  e.  ( 0..^ ( O `  A ) )  |  ( y (.g `  G
) A )  e. 
{ x  e.  ( K `  { A } )  |  ( O `  x )  =  ( O `  A ) } }
)
4 odhash.x . . . . 5  |-  X  =  ( Base `  G
)
5 eqid 2358 . . . . 5  |-  (.g `  G
)  =  (.g `  G
)
6 odhash.o . . . . 5  |-  O  =  ( od `  G
)
7 odhash.k . . . . 5  |-  K  =  (mrCls `  (SubGrp `  G
) )
84, 5, 6, 7odf1o2 14983 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  e.  NN )  -> 
( y  e.  ( 0..^ ( O `  A ) )  |->  ( y (.g `  G ) A ) ) : ( 0..^ ( O `  A ) ) -1-1-onto-> ( K `
 { A }
) )
9 f1ocnv 5568 . . . 4  |-  ( ( y  e.  ( 0..^ ( O `  A
) )  |->  ( y (.g `  G ) A ) ) : ( 0..^ ( O `  A ) ) -1-1-onto-> ( K `
 { A }
)  ->  `' (
y  e.  ( 0..^ ( O `  A
) )  |->  ( y (.g `  G ) A ) ) : ( K `  { A } ) -1-1-onto-> ( 0..^ ( O `
 A ) ) )
10 f1of1 5554 . . . 4  |-  ( `' ( y  e.  ( 0..^ ( O `  A ) )  |->  ( y (.g `  G ) A ) ) : ( K `  { A } ) -1-1-onto-> ( 0..^ ( O `
 A ) )  ->  `' ( y  e.  ( 0..^ ( O `  A ) )  |->  ( y (.g `  G ) A ) ) : ( K `
 { A }
) -1-1-> ( 0..^ ( O `  A ) ) )
118, 9, 103syl 18 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  e.  NN )  ->  `' ( y  e.  ( 0..^ ( O `
 A ) ) 
|->  ( y (.g `  G
) A ) ) : ( K `  { A } ) -1-1-> ( 0..^ ( O `  A ) ) )
12 ssrab2 3334 . . 3  |-  { x  e.  ( K `  { A } )  |  ( O `  x )  =  ( O `  A ) }  C_  ( K `  { A } )
13 fvex 5622 . . . . . 6  |-  ( K `
 { A }
)  e.  _V
1413rabex 4246 . . . . 5  |-  { x  e.  ( K `  { A } )  |  ( O `  x )  =  ( O `  A ) }  e.  _V
1514f1imaen 7011 . . . 4  |-  ( ( `' ( y  e.  ( 0..^ ( O `
 A ) ) 
|->  ( y (.g `  G
) A ) ) : ( K `  { A } ) -1-1-> ( 0..^ ( O `  A ) )  /\  { x  e.  ( K `
 { A }
)  |  ( O `
 x )  =  ( O `  A
) }  C_  ( K `  { A } ) )  -> 
( `' ( y  e.  ( 0..^ ( O `  A ) )  |->  ( y (.g `  G ) A ) ) " { x  e.  ( K `  { A } )  |  ( O `  x )  =  ( O `  A ) } ) 
~~  { x  e.  ( K `  { A } )  |  ( O `  x )  =  ( O `  A ) } )
16 hasheni 11440 . . . 4  |-  ( ( `' ( y  e.  ( 0..^ ( O `
 A ) ) 
|->  ( y (.g `  G
) A ) )
" { x  e.  ( K `  { A } )  |  ( O `  x )  =  ( O `  A ) } ) 
~~  { x  e.  ( K `  { A } )  |  ( O `  x )  =  ( O `  A ) }  ->  (
# `  ( `' ( y  e.  ( 0..^ ( O `  A ) )  |->  ( y (.g `  G ) A ) ) " {
x  e.  ( K `
 { A }
)  |  ( O `
 x )  =  ( O `  A
) } ) )  =  ( # `  {
x  e.  ( K `
 { A }
)  |  ( O `
 x )  =  ( O `  A
) } ) )
1715, 16syl 15 . . 3  |-  ( ( `' ( y  e.  ( 0..^ ( O `
 A ) ) 
|->  ( y (.g `  G
) A ) ) : ( K `  { A } ) -1-1-> ( 0..^ ( O `  A ) )  /\  { x  e.  ( K `
 { A }
)  |  ( O `
 x )  =  ( O `  A
) }  C_  ( K `  { A } ) )  -> 
( # `  ( `' ( y  e.  ( 0..^ ( O `  A ) )  |->  ( y (.g `  G ) A ) ) " {
x  e.  ( K `
 { A }
)  |  ( O `
 x )  =  ( O `  A
) } ) )  =  ( # `  {
x  e.  ( K `
 { A }
)  |  ( O `
 x )  =  ( O `  A
) } ) )
1811, 12, 17sylancl 643 . 2  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  e.  NN )  -> 
( # `  ( `' ( y  e.  ( 0..^ ( O `  A ) )  |->  ( y (.g `  G ) A ) ) " {
x  e.  ( K `
 { A }
)  |  ( O `
 x )  =  ( O `  A
) } ) )  =  ( # `  {
x  e.  ( K `
 { A }
)  |  ( O `
 x )  =  ( O `  A
) } ) )
19 simpl1 958 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  e.  NN )  /\  y  e.  ( 0..^ ( O `  A
) ) )  ->  G  e.  Grp )
20 simpl2 959 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  e.  NN )  /\  y  e.  ( 0..^ ( O `  A
) ) )  ->  A  e.  X )
21 elfzoelz 10967 . . . . . . . . 9  |-  ( y  e.  ( 0..^ ( O `  A ) )  ->  y  e.  ZZ )
2221adantl 452 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  e.  NN )  /\  y  e.  ( 0..^ ( O `  A
) ) )  -> 
y  e.  ZZ )
234, 5, 7cycsubg2cl 14754 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  y  e.  ZZ )  ->  ( y (.g `  G
) A )  e.  ( K `  { A } ) )
2419, 20, 22, 23syl3anc 1182 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  e.  NN )  /\  y  e.  ( 0..^ ( O `  A
) ) )  -> 
( y (.g `  G
) A )  e.  ( K `  { A } ) )
25 fveq2 5608 . . . . . . . . 9  |-  ( x  =  ( y (.g `  G ) A )  ->  ( O `  x )  =  ( O `  ( y (.g `  G ) A ) ) )
2625eqeq1d 2366 . . . . . . . 8  |-  ( x  =  ( y (.g `  G ) A )  ->  ( ( O `
 x )  =  ( O `  A
)  <->  ( O `  ( y (.g `  G
) A ) )  =  ( O `  A ) ) )
2726elrab3 3000 . . . . . . 7  |-  ( ( y (.g `  G ) A )  e.  ( K `
 { A }
)  ->  ( (
y (.g `  G ) A )  e.  { x  e.  ( K `  { A } )  |  ( O `  x )  =  ( O `  A ) }  <->  ( O `  ( y (.g `  G
) A ) )  =  ( O `  A ) ) )
2824, 27syl 15 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  e.  NN )  /\  y  e.  ( 0..^ ( O `  A
) ) )  -> 
( ( y (.g `  G ) A )  e.  { x  e.  ( K `  { A } )  |  ( O `  x )  =  ( O `  A ) }  <->  ( O `  ( y (.g `  G
) A ) )  =  ( O `  A ) ) )
29 simpl3 960 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  e.  NN )  /\  y  e.  ( 0..^ ( O `  A
) ) )  -> 
( O `  A
)  e.  NN )
304, 6, 5odmulgeq 14969 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  y  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( O `
 ( y (.g `  G ) A ) )  =  ( O `
 A )  <->  ( y  gcd  ( O `  A
) )  =  1 ) )
3119, 20, 22, 29, 30syl31anc 1185 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  e.  NN )  /\  y  e.  ( 0..^ ( O `  A
) ) )  -> 
( ( O `  ( y (.g `  G
) A ) )  =  ( O `  A )  <->  ( y  gcd  ( O `  A
) )  =  1 ) )
3228, 31bitrd 244 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  e.  NN )  /\  y  e.  ( 0..^ ( O `  A
) ) )  -> 
( ( y (.g `  G ) A )  e.  { x  e.  ( K `  { A } )  |  ( O `  x )  =  ( O `  A ) }  <->  ( y  gcd  ( O `  A
) )  =  1 ) )
3332rabbidva 2855 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  e.  NN )  ->  { y  e.  ( 0..^ ( O `  A ) )  |  ( y (.g `  G
) A )  e. 
{ x  e.  ( K `  { A } )  |  ( O `  x )  =  ( O `  A ) } }  =  { y  e.  ( 0..^ ( O `  A ) )  |  ( y  gcd  ( O `  A )
)  =  1 } )
3433fveq2d 5612 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  e.  NN )  -> 
( # `  { y  e.  ( 0..^ ( O `  A ) )  |  ( y (.g `  G ) A )  e.  { x  e.  ( K `  { A } )  |  ( O `  x )  =  ( O `  A ) } }
)  =  ( # `  { y  e.  ( 0..^ ( O `  A ) )  |  ( y  gcd  ( O `  A )
)  =  1 } ) )
35 dfphi2 12939 . . . 4  |-  ( ( O `  A )  e.  NN  ->  ( phi `  ( O `  A ) )  =  ( # `  {
y  e.  ( 0..^ ( O `  A
) )  |  ( y  gcd  ( O `
 A ) )  =  1 } ) )
36353ad2ant3 978 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  e.  NN )  -> 
( phi `  ( O `  A )
)  =  ( # `  { y  e.  ( 0..^ ( O `  A ) )  |  ( y  gcd  ( O `  A )
)  =  1 } ) )
3734, 36eqtr4d 2393 . 2  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  e.  NN )  -> 
( # `  { y  e.  ( 0..^ ( O `  A ) )  |  ( y (.g `  G ) A )  e.  { x  e.  ( K `  { A } )  |  ( O `  x )  =  ( O `  A ) } }
)  =  ( phi `  ( O `  A
) ) )
383, 18, 373eqtr3a 2414 1  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  e.  NN )  -> 
( # `  { x  e.  ( K `  { A } )  |  ( O `  x )  =  ( O `  A ) } )  =  ( phi `  ( O `  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710   {crab 2623    C_ wss 3228   {csn 3716   class class class wbr 4104    e. cmpt 4158   `'ccnv 4770   "cima 4774   -1-1->wf1 5334   -1-1-onto->wf1o 5336   ` cfv 5337  (class class class)co 5945    ~~ cen 6948   0cc0 8827   1c1 8828   NNcn 9836   ZZcz 10116  ..^cfzo 10962   #chash 11430    gcd cgcd 12782   phicphi 12929   Basecbs 13245  mrClscmrc 13584   Grpcgrp 14461  .gcmg 14465  SubGrpcsubg 14714   odcod 14939
This theorem is referenced by:  proot1hash  26842
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-iin 3989  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-oadd 6570  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-sup 7284  df-card 7662  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-2 9894  df-3 9895  df-n0 10058  df-z 10117  df-uz 10323  df-rp 10447  df-fz 10875  df-fzo 10963  df-fl 11017  df-mod 11066  df-seq 11139  df-exp 11198  df-hash 11431  df-cj 11680  df-re 11681  df-im 11682  df-sqr 11816  df-abs 11817  df-dvds 12629  df-gcd 12783  df-phi 12931  df-ndx 13248  df-slot 13249  df-base 13250  df-sets 13251  df-ress 13252  df-plusg 13318  df-0g 13503  df-mre 13587  df-mrc 13588  df-acs 13590  df-mnd 14466  df-submnd 14515  df-grp 14588  df-minusg 14589  df-sbg 14590  df-mulg 14591  df-subg 14717  df-od 14943
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